The fastest way I've found to level up Egg Morty (or any low-level Morty) is to bring them along, then only use them when you're up against a Moon/Sun/Black Hole Morty, and let them take the Implode for the team. The higher the disparity in levels between the trainer's (or wild) Morty's level and your Egg (or any other) Morty's levels, the more points they'll get, and the more levels they'll grow. I've had Mortys go up as much as a dozen levels in one hit using this method, and it's now my go to way of levelling up a very low-level Morty...just know that once you hit level 20 (or higher), Egg Morty will only turn into The One True Morty by using a level-up mega seed.
You're a lonely, insignificant speck on a has-been planet orbited by a cold, indifferent sun!